BERNS ON HEARTSCopyright 2002 Gerald M. Berns 1995-2002IntroductionThe game of Hearts (also, and more descriptively, known as Black Lady) is sosimple! The dealer deals the deck one card at a time to four players. Fromthe 13-card hand, on three of every four deals, each player picks three cardsto pass to an opponent. The first pass is to the player on the left; onsuccessive deals the pass is to the right, then across, and then no pass (a"Hold" hand). Then the passing cycle repeats. Play begins with the player atthe dealer's left leading a card (a common variation: the player who has 2 ofclubs (2C) leads it). The card led establishes the trick's suit, and eachplayer in turn plays a card in that suit, if she has one; if not, a player maysluff any card. The player whose card is highest in the suit led wins thetrick and leads the next trick. However, a heart may not be led until heartsare "broken" someone sluffs a heart or plays the queen of spades (QS) or thetrick-leader's hand contains only hearts. Hearts is a negative scoring game:points captured are (usually) bad, not good. Capturing each heart counts onepoint against the capturer, and capturing QS counts 13 points against.However, there is a hitch. If a player captures all 26 points, she "Shootsthe Moon"! For this achievement she receives no points and her threeopponents each receive 26 points; however, if adding 26 points to each of heropponents scores does not put her in the lead, the Shooter subtracts 26 pointsfrom her score. The game ends when a player's score reaches or exceeds 100,and the player with the lowest score wins (ties are allowed). These are therules and variations that Killer Hearts supports (some players use othervariations).Hearts is so simple to understand and play that it may seem almost childlike.Consider that there is no bidding, no trump, no wild card, no melding, nocollecting, no partner with whom to communicate and coordinate, no doubt aboutwhether a particular card has been dealt, and no arcane scoring system.Unlike Bridge, it puts no premium on memorizing conventions and systems.Unlike Setback, it puts little premium on courage and daring. Unlike Checkersand Chess, it puts no premium on depth of vision down a move tree. Moreover,in three of every four hands a player can pass away her three most vexingcards.Despite its simple rules, Hearts is a surprisingly complex game. Toappreciate its complexity, first consider the goals of playing a hand inseveral other card games:Bridge: (Bid and) win the maximum number of tricksSetback: (Bid and) win the maximum number of pointsCasino: Win the maximum number of pointsCanasta: Make the maximum number of meldsThe goals of these games are straightforward and invariant. Now consider thegoals of playing a hand of Hearts, most important first.If currently low-scorer: Maintain winning position andPlay so that an opponent goes outExpand scoring margin over second-place playerPlay so that high-scorer takes maximum number of pointsPlay so that second-low player takes maximum number of pointsIf not currently low-scorer:Become low-scorerProtect opponent with very high score from going outPrevent Moon Shoot that wins gamePrevent Moon Shoot that produces new low-scorerPlay so that low-scorer takes maximum number of pointsWin the minimum number of pointsSome may object to the presence, absence, or order of various goals. Othersmay object that some goals should be further caveated, that they do not applywhen a player's score is very high. But that there are multiple,score-sensitive goals is undeniable. Goals run the gamut from straightforwardto subtle. They can vary from hand to hand and even within a single Heartshand! With a multiplicity of shifting goals comes complexity.Hearts has surprisingly subtle strategies in keeping with its many goals.They give the game a captivating charm. Then, too, there is the "catastrophe"(as in Catastrophe Theory) at the center of the game. If you succeed too wellin ducking points, an opponent may collect all of them (Shoot the Moon), whichmay cost you 26 points. If you attempt to Shoot the Moon and almost succeed,your score can increase by as much as 25 points instead of decreasing by 26.That is, if you are too good at avoiding points or not good enough at Shootingthe Moon, look out!The game is called Hearts, but it is really all about handling QS. In thispaper I explore some of the nuances and strategies that make playing Hearts sochallenging and so much fun. The PassThe three-card pass enables you to improve the hand that random chance dealtyou. The question is: Which are the three best cards to pass? (This is oneof the most important questions that the game asks, because studies withKiller Hearts show that, in general, a larger change in performance resultsfrom changing passing strategy than from changing playing strategy.)High cards take tricks, taking tricks captures points, and points countagainst you. So your first thought might be to pass your highest cards. Butwait! Passing your three highest cards to a skilled opponent may enable herto Shoot the Moon! Conversely, passing three low cards may enable an opponentto avoid taking tricks and cause you to take tricks and points. What to do?Many players (and Hearts-playing computer programs) are so concerned about anopponent Shooting the Moon that they usually pass at least one beatable heart.The idea is that the recipient cannot Shoot the Moon with a beatable heart inher hand, because the passer or someone else will take the trick when sheplays it. This thinking rests on three premises:* Skilled players frequently Shoot the Moon* The pass receiver does not have many heart winners (if she does, the passed heart will not hurt her and may actually help: it is one less heart that the Shooter needs to pull, and it may help her avoid taking tricks if she decides not to Shoot)* Someone else will take a trick on which a beatable heart is playedBut attempting to Shoot the Moon presents a sizable risk, because a close missis very expensive. So, even skilled players approach it with caution.Capturing all the points except the last heart results in a 51 point swingagainst the would-be Shooter: 25 points against versus 26 points for.Furthermore, even though a heart trick only counts a few points against thetaker, there is no guarantee that someone will take a heart-bearing trick.Experienced players often duck such tricks (for various reasons, including"Don't worry, someone else will take it!"). To guarantee that the recipientcannot Shoot the Moon, there is only one way to pass: pass a heart that youcan beat and retain AH, KxH (KH and a lower heart), QxxH, JxxxH, etc. Playyour stopper when the heart you passed is high on a heart trick and the pointshave not yet been split.Of the three premises, only the second is consistently encountered whenplaying with people.Shooting the Moon is a relatively infrequent event: against expertcompetition, even the best players cannot do it more often than about onceevery 32 hands, on average. So why habitually use your pass to protectyourself against the pass recipient's Shooting the Moon, especially when thatprotection is weak? You can purge your hand of only three cards. Why spend athird of your pass in mechanically passing a heart? A policy of passing aheart means that often another problematic card must stay in your hand.Selecting the three best cards to pass (which may also be thought of as thethree worst cards to keep) is an intelligent process that must consider manyfactors, including scores and the passer's degree of risk aversion. Severalfactors are mutually contradictory, so it is a subjective process and expertsdo not always agree on the result. Nevertheless, there are some basics toconsider. The first concerns the all-important spade suit.Suppose you are dealt QS (a.k.a. "the Lady," as well as many not-so-politenames). The numero uno question is: Should you keep it or pass it? If youpass it, your opponents cannot repeatedly lead spades and force you to play itand possibly take it (because only AKS have more capturing strength). Butpassing QS means giving up control over who captures her. Furthermore,passing QS may cause the recipient to view your deed as a social act on a parwith spitting on her sandwich. She may react harshly, looking to return theunwelcome gift to you post haste.Keeping QS can affect your pass. Suppose you also hold AQ94C. If you do notkeep QS, you might pass AQC leaving the 94C doubleton. But, keeping QS, youmight instead pass 94C leaving the AQC doubleton. When play begins, you takethe first club trick with the AC and lead back the QC. Now, possibly as earlyas trick 3, you have a club void through which you may be able to sluff yourQS! Furthermore, if QC wins, you retain the lead and can continue to purgeyour winners. In so doing you may even gather a heart, killing youropponents' Shoot hopes. Holding QS in your hand gives you immunity fromunexpectedly winning her in a trick.Controlling QS means more than simply sluffing it at the first opportunity.Assuming that you are playing Hearts to win and not just for the fun ofparticipating, and that you have enough spades to protect your QS, controllingQS means planning to give it to a particular player. Who your target shouldbe is entirely score-dependent. If you are not the low scorer (i.e., notcurrently winning the game) and the game is well along, you target thatplayer, trying to make her the ex-low scorer. Although giving QS to anyplayer whose score is lower than yours may appeal, the hard fact is that youcannot win until the low scorer is dislodged. On the other hand, if you arenot winning the game you must not give QS to a player who is at the brink ofelimination, for eliminating her eliminates you!If you are the low scorer, you have more potential targets: the high scorer topush her score to 100, a close competitor to increase your winning margin,and, in a crunch, anyone but you.It nets out like this: You should always keep and thereby control QS if youcan tolerate the risk.When is the benefit of holding QS worth the risk? There is more to thisquestion than merely assessing how many spades your opponents hold and theireagerness to pull your QS out with them. Remember that one may sluff any cardwhen void in the suit led. Therefore, the risk analysis reduces to a singlecompound question: Can your opponents lead spades often enough to pull QS outof your hand before someone leads a suit in which you are void in a trick thatone of your targets will probably take?Now we are tickling the essence of the pass. You can tolerate holding QS withfewer protective spades if you have a void, and the earlier you have the voidthe better.The concept of a suit void is larger than a discussion of passing QS. If youpass to make a void, you are in a position (unless the incoming pass fills thevoid) to sluff any card whenever that suit is led. This is an excellentdefensive position. If you think someone (not just your pass recipient) istrying to Shoot the Moon, you can sluff a heart on another player's trick. Oryou can use the void to sluff problem cards or to create additional voids.And, of course, you can sluff QS. Defensive Hearts, wherein you ensure thatpoints are split among at least two players, often depends on the managementof voids.# Suit Distribution Frequency (%)1 13 0.002 12 1 0.003 11 2 0.004 11 1 1 0.005 10 3 0.006 10 2 1 0.007 10 1 1 1 0.008 9 4 0.009 9 3 1 0.0110 9 2 2 0.0011 9 2 1 1 0.0112 8 5 0.0013 8 4 1 0.0414 8 3 2 0.1015 8 3 1 1 0.1116 8 2 2 1 0.1917 7 6 0.0018 7 5 1 0.1019 7 4 2 0.3620 7 4 1 1 0.3921 7 3 3 0.2622 7 3 2 1 1.8823 7 2 2 2 0.5124 6 6 1 0.0725 6 5 2 0.6526 6 5 1 1 0.7027 6 4 3 1.3228 6 4 2 1 4.7029 6 3 3 1 3.4430 6 3 2 2 5.6431 5 5 3 0.8932 5 5 2 1 3.1733 5 4 4 1.2434 5 4 3 1 12.9335 5 4 2 2 10.5736 5 3 3 2 15.5137 4 4 4 1 2.9938 4 4 3 2 21.5539 4 3 3 3 10.53Table 1. The 13-Card Hand's 39 Suit Distributions and Frequency of Occurrence.Only 39 suit distributions are possible in a 13-card hand. Table 1 showsthem, along with their frequency of occurrence (calculations courtesy of BrianM. Berns, my son). Suit distribution #1 in Table 1 is all cards in a singlesuit, which occurs in less than 0.01% of hands, and #39 is the "normal"distribution of one four-card suit and three three-card suits, which occurs in10.53% of all hands. Note, however, that four distributions are morefrequent, and that just five distributions (#34, 35, 36, 38, 39) account fornearly 75% (71.09%) of the hands.Table 1 holds for hold hands. Hold hands occur only 25% of the time, so theytend to get overlooked. That is unfortunate. Table 1's distributions andfrequencies of occurrence can be used to answer an often-asked question abouthold hands. If a player is not dealt QS, how many times can she lead the samenon-spade suit without serious risk that the person holding QS is void in thesuit (the implication being that, if the QS-owner is void in the suit, shewill unload her cargo)? Table 1 shows that hands with at least one voidcomprise 5.04% of all hands, and that hands with a void or singleton comprise35.55% of all hands. Recognizing that the short suit in question is spades25% of the time and that a player is dealt QS 25% of the time, we find thatthe player dealt QS is also dealt a void 0.95% of the time, and is dealt avoid or singleton that is not spades 6.67% of the time. So, in a hold hand aplayer can lead the same non-spade suit twice with less than 7% risk that theQS-owner is void in the suit. Further reducing the risk are theconsiderations that someone other than the trick-leader will win the trick,and that the QS-owner may not be interested in putting QS on the trick-winner.Experience with Killer Hearts supports this conclusion.Table 1 shows, somewhat surprisingly, that a three-card pass to make at leastone void can be made from every hand! A pass to leave two voids can be madein 29 of the 39 possible hands (which collectively occur in 14.95% of allhands, or about once in every seven hands). This is not to mention thesingletons and doubletons that passing can produce. The three-card pass canwreak havoc with suit distributions! The trick in Hearts is to take maximumadvantage of skewed distributions in general and voids in particular.The only spades frequently passed are AKQ. AKS are passed because they cancapture QS. Players dealt one or both of AKS have a similar keep-or-passcalculus as with QS, and they often pass one or both. So, if you pass QS, youmay receive AS or KS in the mail! However, if you keep QS, receiving AS or KSadds to your spade holding and is a significant asset, not a burden.There is another consideration when you are dealt QS in a short spade suit, ifyou are playing against experts. If you are not the low scorer, and if youare passing to someone who is also not the low scorer, she may be in a betterposition to pin QS onto the low scorer (e.g., she may have a longer spade suitor a non-spade void, or she may sit last on a trick when the low scorerleads). In this case, and only in this case, you should consider passing herQS and at least one other spade to mitigate her risk and soften the pass. Butif she is the low scorer, or you are not sure of her skill and experience, oryou are passing to a computer player (other than Killer Hearts' Better or BestPlayer), I do not recommend such a pass.So, to pass QS or not to pass? I suggest the following guidelines fordefensive hands. Pass QS if the hand dealt to you contains:* Two or fewer other spades* Three other spades, the pass is to the right, and the player to your right is not low-scorer; in this case you always play immediately after the QS-owner, which means that you can safely play your big spades when spades are ledOtherwise, keep QS. Remember that these are just guidelines. Bear in mindthat the pass coming your way may fill your voids! If you expect your targetto be so elusive that you may have to hold QS for many tricks, you may needmore spades. Make a special effort not to pass QS to the low scorer, for sheis the last person you want controlling QS.Most people who pass a heart pick a beatable one, as we have discussed.However, do not be afraid to pass your highest heart. The probability is lowthat the pass-recipient has enough other hearts without losers to mount aserious Shoot attempt. Note, however, that passing AH by itself does notdevalue the recipient's heart void if she plans to Shoot.That a heart cannot be led at will makes a heart void less effectivedefensively than all other voids. Because a heart is usually not led untilafter all other suits have been played at least once, you should not expect aheart void to be useful until midway through a hand. On the other hand, aspade void is even more valuable than a diamond or club void, because spadesare usually led early and often. Of course, passing to achieve a spade voidis most risky, as the pass coming your way may contain some of AKQS! Onlywhen you are dealt a spade singleton, or a doubleton or tripleton from amongAKQS, should you consider passing all your spades.My rule of thumb for all modes of play is that a void and a doubleton arebetter than two singletons, a void and a tripleton are better than a singletonand a doubleton, and a singleton and a tripleton are better than twodoubletons. However, I do not recommend passing a deuce or a trey, unlessyour need is urgent (e.g., you have decided to keep QS with few spades).Table 2 lists some personal passing preferences for defensive hands (asopposed to Shooting hands) that do not have QS. Before Pass Preferred Result Not Preferred(to pass 2) T86 KQ53 6 KQ53 T86 53(to pass 2) AJ QJ4 void QJ4 AJ 4(to pass 2) 862 Q8 862 void 2 Q8(to pass 2) 72 AT5 2 T5 void AT5; 72 5(to pass 2) A32 KJ53 32 J53 A32 53; 2 KJ53(to pass 1) KJ7H J7H K7HTable 2. Personal Passing Preferences for Defensive Hands without QS.Until now we have considered the pass only as a defensive operation. This isappropriate because Hearts is essentially a defensive game in which playersnormally try to avoid taking points. Shooting the Moon, however, is anoffensive operation. It is the only playing mode in which one sets out to getmore than one point.Even seasoned players approach Shooting the Moon with trepidation, because ofthe associated risk. Inexperienced players tend to believe that the only wayto Shoot the Moon while keeping the risk tolerable is with a hand having along, strong run of hearts. This looks like the easiest and surest way tocapture all the hearts, and, should they fail to get QS, their flirtation withShooting only costs 13 points. But a solid high heart run occursinfrequently. Experienced players do not turn away from such a heart run, butthey do not wait for one, either. A subtler approach to Shooting the Moon is,perversely, to avoid holding hearts entirely. With a hand that contains oneor more long runs of high cards (and, preferably, also low cards, so that youcan avoid taking tricks if the Shoot goes awry), consider voiding yourself inhearts if you are dealt three or fewer heart losers. Such a hand occurs muchmore often than a long, strong heart suit. If a pass fills your heart void,then you have only those hearts to handle. If you play a heart and lose it,then cancel your Shoot. On the other hand, your Moon Shoot may be launched ifit wins.The word to remember when deciding your best passing strategy is: Voids!Playing the HandAs mentioned earlier, the name of the game may be Hearts, but the game is allabout QS! To a very large and perhaps surprising degree, hearts themselvesplay a small role. That each heart captured costs one point is interestingbut usually unimportant. Hearts games revolve around QS, because its value is13 points. In the large majority of Hearts games, the winner is the playerwho takes QS least often, and the last-place finisher is the player who takesher most often. If the rules changed so that a heart had no point value,Hearts would be played almost the same way.Therefore, to win at Hearts is simple: Avoid taking QS!Unfortunately, trying to avoid QS is what everyone else is doing, too. Whatyou must expect and accept is that, no matter how well you play, the BlackLady usually embraces every player more than once in every game. No oneavoids her for too long. Winners at Hearts just avoid her more often thanothers.Hearts may be a simple game, but it is not simple-minded. The way I (and,therefore, Killer Hearts) view the game, it has eight discrete modes of play,some not so obvious:Playing as low-scorerPlaying to lie lowPlaying to "get" the low-scorerPlaying to Shoot the MoonPlaying while equivocating about Shooting the MoonPlaying to stop a Moon ShootPlaying to help a high-scorer Shoot the MoonPlaying to protect a high-scorerThese modes are primarily dependent on your score relative to the scores ofyour opponents. Other Hearts-playing programs that I have examined do notseem to consider score-dependency an important aspect of the game.Playing a Hearts hand consists of answering one of three questions on every trick:What shall I lead?Shall I follow high or low?What shall I sluff?Each playing mode may require a different answer. To excel at Hearts, youmust be familiar with all modes.Playing Defensively: Who Has the Queen?To win at Hearts, one must avoid taking QS as much as possible. This becomesa much easier task if you know where she sits. However, in most cases whenyou do not hold QS yourself and have not passed her, determining who holds QSrequires slicing through considerable ambiguity. The ability to infer whereQS sits is a major factor in separating excellent Hearts players fromcannon-fodder. If you have figured out who has QS, your problem is eased.You can play as high as you like after she has played. But, if you win thetrick, the problem of selecting a safe lead remains.The best situation is to have QS yourself and let your opponents worry aboutwho has her. And you know where she is if you passed her. If you are passedKS or AS, the passer did not have QS before she received her pass, because aperson holding QS keeps all her spades as protection. The player sluffing ona trick that the low-scorer is winning (or is likely to win) does not have QS(or is trying to Shoot the Moon). A player leading a sequence of unusuallyhigh cards probably has QS, or is Shooting and wants her. A player followingwith unusually high cards may have QS or believe that the person with QS hasalready played (a case in which you learn by observing an opponent's play). Aplayer leading the AS or KS has her, is Shooting, is bluffing a Shoot, or haslost her mind completely.Playing Defensively: General ConsiderationsNew players are acutely aware of this logical progression:High cards win tricksWinning tricks catch QSA common neophyte strategy is: never take a trick! This can work well earlyin a hand, but by the end of the hand the newcomer's low cards have all beenplayed, and guess what happens? You do not want to take more tricks thannecessary, but every player usually takes tricks in every hand. However, thebetter players take tricks early in the hand, hoping that the holder of QS isnot yet void. The best situation in which to take a trick when it is late ina hand is when you play last and have an "out-card" (one that will lose atrick) available to lead.If you do not know who has QS, the earlier you play on a non-spade trick, thegreater the risk of playing high, and leading high is the riskiest. Theearlier you play on a trick, the more players behind you who might have avoid, QS, and the inclination to drop her on you. If a suit in which you havefive or more cards is led for the first time, do not play high unless you sitlast. With the possibility of sluffing it through a void, you may never haveto lead or follow with the highest card in this suit. On the other hand, ifyou have four or fewer cards in the suit and poor chances to develop an earlyvoid elsewhere to sluff winners in this suit, you may have to swallow hard andgo high on a trick. Much better to go high early in a hand than late.If you do not hold Qs or know where she sits, selecting a card to leadrequires care, because you cannot know in which suits the player holding QS isvoid. Dastardly players have even been known to drop QS on the first trick(rules permitting)! There is no foolproof defense against this except a]mindlessly leading spades, b] leading the lowest unplayed card in a live suit(one in which at least one opponent is not void), or 3] standing in good withthe Lord. Hold hands have more reliable distributions. A hold hand with avoid occurs in just 5.18% of the cases (see Table 1). So, an opponent isdealt a Hold hand with QS and a diamond or club void with a frequency of just0.86%, which is once per 116 hands. Here are guidelines for leading when youdo not hold QS and it has not yet been played:* Lead spades if you know it is safe (i.e., it will not harm the wrong player)* Never lead a high card in a long suit if it is long in your hand, it is short in others If you do not know who holds QS:* Never lead high in the suit in which you were passed three cards; your passer may now be void and have a reward for your obliviousness* Never lead the suit in which the player on your left passed you two or more cards, for she may now be void and have a similar reminderThe riskiest holding in the game of Hearts is a short spade suit containingAS, KS, or QS. Next risky is a non-spade suit of four or more cards in whichthe lowest is a 6 or higher and you do not hold QS. There are four cardslower than a 6, and you should prepare to see them at the most inopportunetime! With such a holding you need to be lucky and have QS fall early onsomeone else, sluff most of this suit, Shoot the Moon, or prepare yourself tomeet the Lady.If you hold QS your problem is different, especially if your holding canwithhold a siege of spade pulling. There is freedom in holding QS with adecently long spade suit! You alone can play without fear of taking her on anon-spade trick. Freed from the anxiety that afflicts everyone else at thetable, you can play as high as you wish. And you should play high early andoften. Your goal is to build voids and eliminate entries into your hand, sothat you can aim QS at a target with imperviousness. It is much moreimportant to achieve this attack position than it is to mask the fact that youhold QS.Sooner or later every Hearts player confronts this philosophical question deepinto a hand: What the hell do I lead now? "I have this here, li'l ol' 2D. Isit an out-card, or is it, by any chance, the 13th diamond?" It is amazing howmany times a deuce or trey in a dead suit is led late in a hand and is awardedQS! So this concept should be tacked on the inside back wall of every seriousHearts player's skull:*** Count the cards! ***Keep a running count of how many cards have been played in each suit that youhold. For extra credit, count all the suits when you hold QS. This can helpyou put her on your target. Only those blessed with exquisite memory willalso be able to remember which cards the opponents still hold.Memory is the secret weapon of Hearts. Remember the critical things. If youare treasuring a 3C as an out-card, remember whether the 2C has been played.If it has not, your 3C may not be an out-card! If you hold QS, remember ifthe AS or KS has fallen. It sometimes happens that you hold QS and realizethat a target holds the only spade still out, a KS or AS. Few plays in Heartsare more dramatic or intimidating than leading QS and causing the AS or KS todrop (this is something that Hearts-playing programs do far better thanpeople)! Remember the suits in which your opponents are void. The player whosluffs after you played high on an earlier trick in this suit either does nothave QS, was targeting someone else, or was Shooting. So, leading this suitis relatively safe: It may get you out of the lead, and it may cause QS tofall on the low scorer. At worst it should cost no more than a heart or two.A good memory is essential when you have the lead late in a hand and QS hasnot yet fallen. No holding in Hearts is more precious than an out-card latein a hand! The truest out-cards are, by definition, the lowest unplayed cardsin the live suits. Other than a low spade, you have only your memory todetermine if you have an out-card. Never squander an out-card.Hearts offers the opportunity to develop many subtle strategies.Nevertheless, I repeat that everyone will take many turns dancing with theLady! You will be doing well to win a third of your games. Bad hands happen,sometimes with painful regularity. I have been dealt a singleton AS, KS, orQS on a Hold hand more times than mere chance would explain! Do not bellyachewhen it is your turn. And when you drop her on a player whom you have beenchasing all evening, do not gloat. The Hearts player's mantra should be,Gloat not that ye be not gloated over! Cackle now and pick her out of yourear in a moment! Did I mention that good Hearts players have good memories?Playing Mode: Low-ScorerBeing low-scorer means being the focus of everyone else's attention, for theyenvy your position and will do anything to displace you. As a wise persononce observed, you are not paranoid if they really are out to get you. Thisbrings us to the concept sometimes inelegantly referred to as "lying low."The idea is for a player to build a "hole" to hide in and then disappear intoit, pulling the hole in after her and taking no more points.Lying low is largely the art of building voids. A big part of lying low isnot going high (hence the term) on a non-spade trick, unless you are sittinglast, are playing a Hold hand, or the player who holds QS has already playedon the trick. As already discussed, making non-hearts voids is the mostimportant strategy in playing Hearts, and the earlier you have a void, themore useful it is. So, if you have the lead early in the game, considerplaying your singletons and doubletons. Use your void to sluff problem cardsand to create another void. Break hearts, if hearts is a good suit for you(i.e., you have few hearts or many low ones), or you are concerned about anopponent Shooting. Certainly split hearts, if one player has all the points.If you are in trouble holding QS, as low scorer you should dump her at yourfirst opportunity. If you do not hold QS, generally your best strategy is tolead spades whenever you can, unless you are in trouble holding the AS or KS.Your main targets are the high-scorer, to nudge her closer to the finish line,and your nearest competitor, to widen your lead.Playing Mode: Lying LowThis mode of play is appropriate when your score is close to low. Here youkeep your eyes down and try not to be noticed. The theory behind this mode ofplay is that sooner or later the low-scorer, who is everyone's target, willget nailed and you will silently slip into first place.If you have QS, aim her at a needy opponent but not on a player who will goout (reach 100 points) causing you to lose! If you note that the low scorerhas had the opportunity to lead spades but has not, you might suspect that shehas spade trouble. Then you should consider pulling spades, unless the lowscorer sits last on the trick and the AS or KS has not been played (if the lowscorer sits last and you lead spades, you may help her out of a tight spot byenabling her to get rid of a naked AS or KS; or you may inadvertently pullanother opponent's AS or KS, causing her to win the low-scorer's QS).Some tricks are worth coming out of seclusion to take, such as the trick withthe first heart. This guarantees that no one can Shoot the Moon but you.Always take a heart-bearing trick if another player has all the points.Taking a few hearts is usually cheap insurance! Another trick to take earlyis one in a suit in which you have a high-low doubleton. Take high, lead backlow. Presto! a void.Playing Mode: Getting Low-ScorerThe new player dumps QS, the expert aims her. Hearts has only one winner andthree losers - second place does not count! Even though Hearts is a cutthroatgame, experienced players learn to play cooperatively. Late in a game thethree high scoring players form a loosely knit, unofficial, unspoken coalitionwith just one purpose: to damage the low scorer.Being able to lay points on the low-scorer is probably the most important (andrarest) skill in Hearts. If a player cannot do this well, once she fallsbehind she can only hope that the low-scorer self-destructs, that she canovertake low-scorer by Shooting the Moon, or that some another opponent hasthe skill to undo her. Since almost everyone wins less than half the games,every Hearts players spends most of her time in this mode.The essence of playing well in this mode is patience and fortitude. If youhold QS and it is not in immediate danger, you must pass up sluffing QS onother targets that may present themselves and wait for low-scorer to take atrick. Try to lead a card that will cause low-scorer to win the trick. Thismeans paying close attention to the suits in which low-scorer is void andbeing very careful not to lead them. If you cannot lead into low-scorer'shand, perhaps you can lead into the hand of an opponent who sits tolow-scorer's right, so that she can then lead into low-scorer's hand.On a pass-across or pass-right hand, it may be worth considering whether topass QS plus a spade or two to the opponent who sits directly behindlow-scorer. This player has the best chance to christen a trick on whichlow-scorer leads high.Playing Mode: Shooting the MoonShooting the Moon is like jujitsu: it uses your opponents' strengths againstthem. Hearts players learn to avoid points. All a Shooter must do is let theother players do what they do best, duck points and discard winners, whilecollecting all the points herself. I told you Hearts is a simple game!What is essential to Shoot the Moon? Clearly, lots of high cards. Althoughthis can be achieved in all four suits, it is usually easier if you have oneor two long, strong suits. You must have very few losers after you have takensome hearts and the other players are beginning to awaken to your intention.You must have the ability to play coolly while counting cards in a pressuresituation. And, most importantly, you must have the nerve to try.Score should be a factor in your decision. If your score is in thestratosphere and you are 20 or more points behind, you are running out of timeand will probably need to Shoot the Moon to win. Because you are in seriousscoring trouble, it may be risk-appropriate to try to Shoot with a hand thatis not as strong as you would like. If your opponents are serious,experienced players, they will help you, as we have discussed - especially ifyour Shoot's failing will cause them to lose the game! On the other hand, ifyou are winning by a lot, the benefit of Shooting does not equal its riskunless your hand is extremely strong.If you are considering Shooting the Moon, you will probably have to pass awaylosers that you would otherwise cherish. At worst your pass may reveal to oneplayer that you are thinking about Shooting what else is she to make ofreceiving 2D, 3D, and 4D! You must also be able to assimilate whatever passyou receive. If the pass you get kills the Shoot (e.g., 2H, 3H, and 4H intoyour heart void), then cancel the Shoot without delay.Play your losers early. Use losers to draw out your opponents' winners,promoting the power of your other cards. However, you need not lead all yourlosers. That is, you can save a loser or two (especially if QS has not yetbeen played) until the last tricks, forcing your opponents to guess the suitof your weakness. Your losers are considerably strengthened if you also holdQS, for few opponents will risk trying to stop your Shoot at the cost ofwinning QS themselves. Once you commit to Shooting, you must take all thepoints quickly or plan to take all the point-bearing tricks. A long, strongheart suit is easiest on the blood pressure, because you can drain all thehearts in a few pulls.Is it possible to Shoot the Moon on a Hold hand? You bet it is!Playing Mode: Equivocating about Shooting the MoonThe best Shooters hedge their bets by thinking offensively and defensively atonce, before they commit outright to Shooting or decide not to try. Mosthands that produce successful Shoots are not ironclad from the outset. Infact, they may have severe weaknesses that, through skill or good fortune, areovercome. When equivocating, one plays losers on tricks without points,gradually building up the hand's strength in its remaining cards. The skillis in being able to prune losers without having to commit to Shooting andwithout being overly punished for one's boldness.Hands with one or more long suits lend themselves especially well toequivocating. A long suit often has out-cards at the bottom end with which toLie Low if a heart should fall to an opponent.Playing Mode: Stopping a Moon ShootThe defense is usually at a disadvantage when someone tries to Shoot the Moon.Because a serious Shoot attempt generally occurs less than once a game, itsinfrequency can mask its presence. It may take several tricks before theircommon problem dawns on the others. Most Hearts players develop finedefensive skills, because that mode dominates Hearts play. Stopping a Shootrequires taking a trick and seeking a point, an entirely different mindset.The best defense against a Shoot is not the pass, which can only hold oneopponent in check, as we have discussed. Rather, the best defense against aShoot is an early void, which can be effective against all opponents. Asluffed heart has aborted many a Moon Shoot, leaving it smoking on thelaunchpad. There is an interesting logical progression here:1) The earlier a player develops a void, then2) The more tricks the Shooter has to control, then3) The stronger the Shooter's hand must be, then4) The less often that Shoot-worthy hands occur, then5) The less often the Shoots, then6) The less often that opponents anticipate a Shoot, then7) The more likely Shoots are to succeed.As mentioned, skillful players do not wait for a perfect hand to Shoot theMoon. Many a Shoot has developed from a hand that originally containedseveral losers. The problems for the defense are to figure out in which suitis the Shooter's loser, then to beat it by taking at least one point but asfew points as possible.The Shoot succeeds when the Shooter captures all the points. Therefore, aslong as the Shooter has not captured some points, the Shoot may beproblematic. The defensive player must guess correctly in which suit theShooter has a problem and retain a winner in it. Someone else must retain apoint card. Only one player has to guess right if the Shooter's weak suit ishearts. However, because the point cards that defenders most often retain aretheir highest hearts, it is unusual for the Shooter's weak suit to be hearts.But what if QS has not fallen and you hold the AS, KS, or QS. Do you reallywant to stop the Shoot that much?Determining the Shooter's weak suit is a problem that memory helps. Rememberin which suits the Shooter is void, for clearly she has no loser in those.Remember to sluff your dead cards except in hearts, where you might want toretain one or two.Merely taking a trick does not stop a Shoot; it must contain a point card.Playing Mode: Helping High-Scorer Shoot the MoonThe three players who are not low-scorer form a cooperative coalition which isdynamic and unstable. Its members accept that just two things bind themtogether:* Each wants to beat the hell out of the others, and* The low scorer is currently thwarting them allUnspoken is each member's knowledge that she has a better opportunity to winif only the low scorer can be dislodged. It matters not that, if thecoalition is successful, one of its members will be the target of a newcoalition.Is an opponent successfully Shooting the Moon always a bad thing? No. If theShooter's score is more than 26 points higher than yours, it may be better tolet her Shoot than to run the risk of stopping her. Consider also the case inwhich you are not low-scorer, the Shooter's score is near 100 points, and herscore is more than 26 points higher than low-scorer's. A Hearts game is overwhen one player reaches 100 points. It follows, therefore, that a Hearts gamein which no one reaches 100 points continues forever! Recall that asuccessful Shooter may deduct 26 points from her score. A Hearts game inwhich high-scorer pulls back from the brink gives new life to the game, for agame that is still being played is one that anyone can win. Considering thatyou lose if you do not win, a high scorer who decides to Shoot and cannot winthe game by doing so should be able to count on two supporters and one enemy(low-scorer, who wants the game to end now). But even low-scorer will not beinterested in stopping this Shoot if it risks dumping her out of the lead.The Shooter's supporters (who may even include low-scorer) discard theirwinners before these cards can capture tricks. They allow the Shooter tocapture quickly their hearts and QS, if they have her. Remember that Shootingthe Moon does not require taking all the tricks, just all the points.Playing Mode: Protecting High-ScorerPlaying cooperatively is an enlightened broadening of the definition of whatconstitutes one's best self-interest. It reaches a high form when a playerrefrains from dropping QS on a member of the coalition (even second-placeplayer), deciding to wait for the coalition to maneuver into a position whereshe can give it instead to low-scorer. Her fellow coalition members, whounderstand (and expect) that she will restrain herself in playing QS,reciprocate by not leading spades unduly, especially when one knows that theonly spade remaining outside her hand is QS. The challenge is to get the leadand lead low in a suit in which the holder of QS is void and low-scorer holdsa winner. Here, again, memory and the fall of the cards are key.Cooperative play can be painful. Consider a trick on which you play last, andyou are not low-scorer. A 5D lead drops low-scorer's QS. A player whosescore is 88 has to play 7D. You have 4D and JD. If you follow your instinctand duck with 4D, high-scorer takes the trick. Congratulations, you have justshot yourself in the foot, for the game is over and you lose! The only way tokeep the game, and your chances to win it, alive is to groan loudly, play JD,and win QS. Another painful situation is when a protecting player knows thatthe only spade still remaining outside her hand is QS, knows that high-scorerholds QS, and realizes that her hand has no other outcard except in spades.If she leads a spade: game over and she loses! She must lead another card andprobably wind up eating QS! But she will live to play another hand, and wherethere is life there are hope and possibility.It is not unknown for a protected high-scorer to Shoot the Moon and win thegame! Few strategies are risk-free.To watch three excellent Hearts players playing cooperatively, stalking alow-scorer adept at evading, is to watch Hearts at its very best.MusingsHearts has a high chaos factor. The chaos is due to the lack of informationabout other hands, especially concerning skewed suit distributions early inplay and the whereabouts of QS. Contrast this with Checkers and Chess, twogames with zero chaos factor because everything is visible and, therefore,knowable.An inherently high chaos level tends to diminish the effect of skill. Acorollary is that poorer players do better against good players in Hearts thanin many other, less chaotic strategy games. One definition of a poorer playeris one who brings randomness (in the form of lack of knowledge) to a game.The effect of a poorer player's randomness is masked in a game with an alreadyhigh level of chaos. The author of another Hearts-playing computer programhas written me that he was once trounced by three computer players who playedby the rules but picked cards at random. If in Killer Hearts a Best Playerplays three Random Players in a set of 5,000 duplicate games, Best Player willprobably not win them all.Still, playing with better players is more challenging and more fun thanplaying with poorer players, and there is a big difference in the feel of thegame. Skill tends to make order out of chaos. As in other, more seriousrealms, intelligence tends to reduce and can even reverse entropy. It followsthat the skill of a good player makes your own skill more important. And yes,although it may seem strange to say so, it is more satisfying to receive QS bydesign rather than by accident, by aim rather than by dump!Much of the chaos in Hearts is due to the three-card pass, because it sodrastically skews suit distributions. It is interesting, however, that mostexperienced players (including me) favor passing hands over Hold hands. Eachbelieves that she can use the pass to improve her hand, despite the passcoming her way.Where does the skill lie in Hearts? It lies in being able to nail a propertarget with QS. In playing cooperatively, because this prolongs the game,thereby increasing winning opportunities for the three players who are not inthe lead. In determining where QS sits and playing accordingly. In passingwell. In playing well with a bad hand. And in evading skillful efforts tostick you with QS. Shooting the Moon is less a skill than a capacity toaccept risk (or recognize opportunity) and the boldness to exploit thedefensive nature of the game. But being able to stop a Shoot consistently isa precious skill.A Little Math and Other Fun ThingsThe best Hearts player does not always win. In fact, she may not even usuallywin. Over a large number of games between the same players, the best playerwill win the most games (this is what makes her best). But lacking a largenumber of games and mindful that low scores win games, can we say that thebest player is the one who takes the fewest points? Moreover, is a player'sscoring average (her proportion of all points taken) an accurate predictor ofher long term winning average? It might seem reasonable to think so. But thesomewhat surprising answers are that the lowest scoring average may not winthe most games, and that scoring average is not a reliable predictor ofwinning average.Since, in Hearts, the lower the score the better, the proposition that winningaverage is inversely proportional to scoring average is expressed byW1 is proportional to 1/p1where W1 is player one's predicted winning average and p1 is her actualscoring average (the percentage of all points that she takes). If the scoringaverages are firm and knowing thatp1 + p2 + p3 + p4 = 1thenW1 = 1 / (p1 * (1/p1 + 1/p2 + 1/p3 + 1/p4))Now consider Table 3, which contains scores from ten consecutive Hearts games.Player 1 is a person and players 2, 3, and 4 are identical replications ofanother computer game's best player. Clearly player 1 has won many more games(8) than his scoring average predicts (.342), and the automata have won fewer.This trend continued over the much larger set of games from which these tenare extracted. Over 100 consecutive games, player 1's predicted win average,W1, was .316, but his actual winning average was .550. He won 55 gamesinstead of the 32 that his scoring average predicted. That is 23 games and74% more than predicted! What goes on here? Player 1 Player 2 Player 3 Player 4Scores in Game 1 62 101 69 80Scores in Game 2 94 96 88 112Scores in Game 3 64 94 67 113Scores in Game 4 6 118 20 12Scores in Game 5 17 101 65 77Scores in Game 6 102 75 83 78Scores in Game 7 51 62 55 118Scores in Game 8 44 65 92 111Scores in Game 9 34 117 89 98Scores in Game 10 57 68 87 100Total Points 531 897 715 899Scoring Average, p .175 .295 .235 .296Predicted Win Avg., W .342 .202 .254 .202Actual Wins 8 1 1 0Actual Win Average .800 .100 .100 .000Table 3. Data from Ten Consecutive Games.The situation can illuminated by example. Consider a hand in progress inwhich players 1, 2, 3, and 4 have 64, 96, 54, and 111 points respectively.When we peer into this hand, all the hearts have been played and split, player1 holds QS with no entries into his hand, and another player has the lead.Clearly player 1 is about to put QS on someone. Player 1 only wins the gameif he gives QS to player 3, but his scoring average will be the same no matterwho his victim is .189 (64/338) and his predicted W for this one game willonly vary from .297 to .313!Part of what we are seeing is the value of being able to put QS where it willdo the most good. If player 1 merely drops QS at random, he wins fewer gamesthan if he aims QS at an appropriate victim but his scoring average isunaffected by his aim! Player 1's ability to aim QS at the proper targetincreased his winning average and helped enable him to win 23 games that hemight have lost with the identical scoring average.So scoring average is not a reliable predictor of winning average. Yourscoring average in part reflects your ability to evade QS and other points.But that is only the defensive aspect of Hearts. Your ability to win is alsovery much a matter of your offensive ability to aim QS what we may call"queensmanship"! Playing cooperatively also skews the predictive ability ofscoring average away from winning average. A player may take points that shedid not have to, were she not trying to keep the high scorer from going outand, therefore, the game and her chances to win it alive. Furthermore, aplayer who often tries to Shoot tends to have a higher scoring average thanotherwise, but if her Shoot success rate is high enough her winning averagewill increase. Does this mean that a player can win more games than someone whose scoringaverage is significantly lower? Yes! For a reductio ad absurdum case showingthat scoring average is decoupled from winning average, consider these fournew players. Player 5 plays successive games in which she scores 112, 111, 5,112, 111, ... points. In those same games player 6 gets 111, 5, 112, 111, 5,... points, while player 7 receives 5, 112, 111, 5, 112, ... points.Meanwhile, player 8 takes just 6 points in every game. Players 5, 6, and 7have scoring averages of .325 but a winning average of .333. Player 8 has thephenomenally low scoring average of .026, but she never wins a game!In the real world, however, where such scoring sequences are highly unlikely,the best player of a group usually does have the lowest scoring average.However, a somewhat better indicator of quality of play than scoring averageis each player's proportion of the number of points in excess of the winningscore per game (the ratio of a player's average losing margin to all losingmargins). This measure (call it D) takes into account a player's ability toaim QS.If the scores of a game are 106, 64, 34, and 82, then the total number ofpoints in excess of the winning score for all players is 150 (72 + 30 + 0 +48), and D1, D2, D3, and D4 are .480, .200, 0, and .320 for this one game.The relationship between D and winning average looks familiarW1 = 1 / (D1 * (1/D1 + 1/D2 + 1/D3 + 1/D4))Using this equation to predict winning averages from the scores shown in Table3 yields .759, .063, .115, and .062 for players 1, 2, 3, and 4. Thiscorrelates quite well with the actual small sample (.800, .100, .100, .000).Having examined the results of this predictor of W over large numbers ofgames, I find that it almost always provides a better estimate of winningaverages than an estimate based on scoring averages. However, it tends tounderestimate higher winning averages and overestimate lower winning averages.It, too, must be considered unreliable.Do players with the same skill level have the same scoring and winningaverages? I have studied at length Killer Hearts and another Hearts-playingprogram, always playing against three replications of each program's bestplayer. Clearly, identical replicants have identical skills. After hundredsof games, I have found that the answer is no. The player playing after me didbest in both scoring and winning averages, the player at my right did worst,and the player sitting across from me fell between them. These data suggestthat the best seat at the Hearts table is behind (to the left of) the best ofthe other three players.For something different, consider this. Can a game end with all four playershaving 100 or more points? Yes. Final scores of 106, 104, 103, 103; 107,104, 104, 101; etc. are possible. To get to the first set, the scores beforethe last deal could have been 93, 99, 99, 99. The highest possible score inHearts is 125, but what is the highest score with all four players at 100 ormore? It is 116 (the scores are 116, 100, 100, 100).Is a four-way tie at a score above 100 possible? No. The total of pointstaken in a game must be a multiple of 26. Consider the possibility of a tieat 104 (4 x 104 = 416 = 16 x 26). To get to this score, the player who tookQS on the last hand could have had no more than 91 points at the start of thehand. That leaves 13 points for the others, but 15 are needed for the otherthree players to have had scores of 99. The next multiple of 26 is 442, whichis not a multiple of 4. Next is 468, but a tie at 117 is not possible; threeplayers would have had scores of at least 109 at the start of the last hand.A player who Shoots the Moon has the option of reducing her score by 26points. Is it possible that a rational player's score can go below zero? No.The only case in which it behooves the successful Shooter to reduce her score,rather than adding 26 points to everyone else s score, is when her scorebefore the Shoot was 26 or more points higher than the low score. So thelowest her score can be before the Shoot is 26 and the lowest it can beafterward is zero.The shortest possible complete Hearts game requires four deals. The shortestgames I have ever seen were over in six (for an example, see game 4 in Table3). The longest game without a successful Shoot consumes 16 deals, but I havenever seen a non-Shoot game exceed 15 deals.The rarest events in a game of Hearts? A player taking no points, a playerscoring 125 points, all four players scoring 100 or more points, a marginbetween first and second place of more than 75 points, and one playersuccessfully Shooting the Moon on three consecutive deals, to name some. (Ihave seen one each of a winner having zero points, a high score of 124 points,three players over 100, 77 points between first and second place, and twostraight Moon Shoots with the same Shooter finally failing on the 13th trickof the third game.)AfterwordHearts is a game in which skill will out eventually, but maybe not today. Forthis reason Hearts tests your capacity to endure adversity, frustration,defeat, and even humiliation. It makes you sweat and wonder what you did tooffend the deity. It makes you think and work at improving your memory. Itteaches: Cooperate or lose! Narcissistic or arrogant people should avoidHearts unless they are very, very skilled. Even so they will still besmashed, as we all are.Hearts is a game that begs not to be taken too seriously. Fun is its mainproduct. Fortunes are not made or lost on it. People do not write songsabout it. There are no systems for beating it. Master points are not awardedfor excelling at it. It is a game that is meant to be played and enjoyed.But having both won and lost, I know that you will enjoy Hearts more if youwin more. It pleases me to think that I may have helped you do that.[Early versions of this paper, which I wrote many years before developingKiller Hearts, helped greatly in designing Killer Hearts.]